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I am generally unsure of what a kernel is in abstract algebra. From my background in linear algebra, I understand that the kernel is the same as the null space (I learned it as the null space).

For example, if I have some homomorphism $f: (\mathbb{R},+) \longrightarrow (\mathbb{C}^*, \cdot)$, how would I go about describing the kernel?

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    $\begingroup$ Kernel is the set of elements of the domain which are mapped to the identity element of the co-domain. $\endgroup$
    – Kavi Rama Murthy
    Nov 12, 2020 at 7:40

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Ther kernel of a (group) homomorphism is the set of all elements that get mapped onto the neutral element.

So $\ker(f)=\{x\in\mathbb{R}: f(x)=1\}$, as $1$ is the neutral element of $\mathbb{C}^\ast$ with regards to multiplication.

How to describe the kernel then depends clearly on the specific homomorphism.

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  • $\begingroup$ So any element that maps to the identity element in the range? Thank you so much! $\endgroup$
    – Georgina Spence
    Nov 12, 2020 at 7:47
  • $\begingroup$ Yes. Also the kernel is never empty, as every homomorphism sends the identity element onto the identity element. But as I said, how to discribe the kernel depends on the homomorphism. As a tip: You should always clearify definitions and notation with your lecture notes. $\endgroup$
    – Cornman
    Nov 12, 2020 at 7:51

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